Tzaban–Paolini–Shelah conjecture on the profinite-group cardinal invariant

Let $G$ be an infinite metrizable profinite group. The cardinal invariant $coperatorname{fm}(G)$ is associated with $G$, and $coperatorname{non}(\mathcal{N})$ denotes the uniformity of the null ideal.

Tzaban–Paolini–Shelah conjecture. For every infinite metrizable profinite group $G$,

$$\mathrm{non}(\mathcal{N}) \leqslant \mathfrak{fm}(G).$$

If true, this would, together with the preceding result, imply that every infinite metrizable profinite group has a non-Haar-measurable subgroup. The conjecture is resolved negatively by the consistency result exhibited in the paper.